2.3 Some asymptotic estimates
We now derive some estimates that will be used throughout the paper. We start from the asymptotic behavior of P
τ
T 1
= n as n → ∞. Let us set gT :=
− log cos π
T
= π
2
2T
2
+ O 1
T
4
, T → ∞ .
2.12 We then have the following
Lemma 2.1. There exist positive constants T , c
1
, c
2
, c
3
, c
4
such that when T T
the following rela- tions hold for every n
∈ 2N: c
1
min {T
3
, n
3 2
} e
−gT n
≤ Pτ
T 1
= n ≤ c
2
min {T
3
, n
3 2
} e
−gT n
, 2.13
c
3
min {T,
p n
} e
−gT n
≤ Pτ
T 1
n ≤ c
4
min {T,
p n
} e
−gT n
. 2.14
The proof of Lemma 2.1 is somewhat technical and is deferred to Appendix B.1. Next we turn to the study of the renewal process
{τ
n
}
n ≥0
, P
δ,T
. It turns out that the law of τ
1
under P
δ,T
is essentially split into two components: the first one at O1, with mass e
δ
, and the second one at OT
3
, with mass 1
− e
δ
although we do not fully prove these results, it is useful to keep them in mind. We start with the following estimates on
P
δ,T
τ
1
= n, which follow quite easily from Lemma 2.1.
Lemma 2.2. There exist positive constants T , c
1
, c
2
, c
3
, c
4
such that when T T
the following rela- tions hold for every m, n
∈ 2N ∪ {+∞} with m n: c
1
min {T
3
, k
3 2
} e
−gT +φδ,T k
≤ P
δ,T
τ
1
= k ≤ c
2
min {T
3
, k
3 2
} e
−gT +φδ,T k
2.15 P
δ,T
m ≤ τ
1
n ≥ c
3
e
−gT +φδ,T m
− e
−gT +φδ,T n
2.16
P
δ,T
τ
1
≥ m ≤ c
4
e
−gT +φδ,T m
. 2.17
Proof. Equation 2.15 is an immediate consequence of equations 2.8 and 2.13. To prove 2.16, we sum the lower bound in 2.15 over k, estimating min
{T
3
, k
3 2
} ≤ T
3
and observing that by 2.3 and 2.12, for every fixed
δ 0, we have as T → ∞ gT +
φδ, T = 2
π
2
e
−δ
− 1 1
T
3
1 + o1 .
2.18 To get 2.17, for m
≤ T
2
there is nothing to prove provided c
4
is large enough, see 2.18, while for m
T
2
it suffices to sum the upper bound in 2.15 over k. Notice that equation 2.15, together with 2.18, shows indeed that the law of
τ
1
has a component at OT
3
, which is approximately geometrically distributed. Other important asymptotic relations
2047
are the following ones: E
δ,T
τ
1
= e
δ
e
−δ
− 1
2
2 π
2
T
3
+ oT
3
, 2.19
E
δ,T
τ
2 1
= e
δ
e
−δ
− 1
3
2 π
4
T
6
+ oT
6
, 2.20
which are proven in Appendix A.2. We stress that these relations, together with equation A.6, imply that, under
P
δ,T
, the time ˆ τ needed to hop from an interface to a neighboring one is of order
T
3
, and this is precisely the reason why the asymptotic behavior of our model has a transition at T
N
≈ N
1 3
, as discussed in the introduction. Finally, we state an estimate on the renewal function P
δ,T
n ∈ τ, which is proven in Appendix B.2.
Proposition 2.3. There exist positive constants T , c
1
, c
2
such that for T T
and for all n ∈ 2N we
have c
1
min {n
3 2
, T
3
} ≤ P
δ,T
n ∈ τ ≤ c
2
min {n
3 2
, T
3
} .
2.21 Note that the large n behavior of 2.21 is consistent with the classical renewal theorem, because
1 E
δ,T
τ
1
≈ T
−3
, by 2.19. One could hope to refine this estimate, e.g., proving that for n ≫ T
3
one has P
δ,T
n ∈ τ = 1 + o1E
δ,T
τ
1
: this would allow strengthening part 1 of Theorem 1.1 to a full convergence in distribution S
N
C
δ
p N
T
N
=⇒ N 0, 1. It is actually possible to do this for n
≫ T
6
, using the ideas and techniques of [7], thus strengthening Theorem 1.1 in the restricted regime T
N
≪ N
1 6
we omit the details.
3 Proof of Theorem 1.1: part 1
We are in the regime when N T
3 N
→ ∞ as N → ∞. The scheme of this proof is actually very similar to the one of the proof of part i of Theorem 2 in [3]. However, more technical difficulties arise
in this context, essentially because, in the depinning case δ 0, the density of contact between
the polymer and the interfaces vanishes as N → ∞, whereas it is strictly positive in the pinning case
δ 0. For this reason, it is necessary to display this proof in detail. Throughout the proof we set v
δ
= 1 − e
δ
2 and k
N
= ⌊NE
δ,T
N
τ
1
⌋. Recalling 2.4 and 2.9, we set Y
T
N
= 0 and Y
T
N
i
= ǫ
T
N
1
+ · · · + ǫ
T
N
i
for i ∈ {1, . . . , L
N ,T
N
}. Plainly, we can write S
N
= Y
T
N
L
N ,TN
· T
N
+ s
N
, with
|s
N
| T
N
. 3.1
In view of equation 2.19, this relation shows that to prove 1.5 we can equivalently replace S
N
C
δ
p N
T
N
with Y
T
N
L
N ,TN
p v
δ
k
N
.
2048
3.1 Step 1
Kategori